# Properties

 Degree 2 Conductor $11^{2}$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Learn more about

## Dirichlet series

 L(s)  = 1 + 2-s + 2·3-s − 4-s + 5-s + 2·6-s − 2·7-s − 3·8-s + 9-s + 10-s − 2·12-s + 13-s − 2·14-s + 2·15-s − 16-s − 5·17-s + 18-s + 6·19-s − 20-s − 4·21-s + 2·23-s − 6·24-s − 4·25-s + 26-s − 4·27-s + 2·28-s + 9·29-s + 2·30-s + ⋯
 L(s)  = 1 + 0.707·2-s + 1.15·3-s − 1/2·4-s + 0.447·5-s + 0.816·6-s − 0.755·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.577·12-s + 0.277·13-s − 0.534·14-s + 0.516·15-s − 1/4·16-s − 1.21·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s − 0.872·21-s + 0.417·23-s − 1.22·24-s − 4/5·25-s + 0.196·26-s − 0.769·27-s + 0.377·28-s + 1.67·29-s + 0.365·30-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$121$$    =    $$11^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{121} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 121,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $1.666156920$ $L(\frac12)$ $\approx$ $1.666156920$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 11$, $F_p(T) = 1 - a_p T + p T^2 .$If $p = 11$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad11 $$1$$
good2 $$1 - T + p T^{2}$$
3 $$1 - 2 T + p T^{2}$$
5 $$1 - T + p T^{2}$$
7 $$1 + 2 T + p T^{2}$$
13 $$1 - T + p T^{2}$$
17 $$1 + 5 T + p T^{2}$$
19 $$1 - 6 T + p T^{2}$$
23 $$1 - 2 T + p T^{2}$$
29 $$1 - 9 T + p T^{2}$$
31 $$1 + 2 T + p T^{2}$$
37 $$1 + 3 T + p T^{2}$$
41 $$1 + 5 T + p T^{2}$$
43 $$1 + p T^{2}$$
47 $$1 - 2 T + p T^{2}$$
53 $$1 - 9 T + p T^{2}$$
59 $$1 - 8 T + p T^{2}$$
61 $$1 - 6 T + p T^{2}$$
67 $$1 - 2 T + p T^{2}$$
71 $$1 - 12 T + p T^{2}$$
73 $$1 + 2 T + p T^{2}$$
79 $$1 + 10 T + p T^{2}$$
83 $$1 - 6 T + p T^{2}$$
89 $$1 + 9 T + p T^{2}$$
97 $$1 + 13 T + p T^{2}$$
show more
show less
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}

## Imaginary part of the first few zeros on the critical line

−19.30198464662403, −18.23029015521226, −17.45129267091964, −15.94960393364486, −15.19519004143324, −14.06941374406207, −13.65039157651841, −12.93331549622801, −11.69772641079316, −9.975450561586109, −9.182457967238805, −8.356936232923193, −6.722214028918875, −5.384155723528973, −3.841779023620474, −2.752740844635733, 2.752740844635733, 3.841779023620474, 5.384155723528973, 6.722214028918875, 8.356936232923193, 9.182457967238805, 9.975450561586109, 11.69772641079316, 12.93331549622801, 13.65039157651841, 14.06941374406207, 15.19519004143324, 15.94960393364486, 17.45129267091964, 18.23029015521226, 19.30198464662403